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Stochastic Search Equilibrium STOCHASTIC SEARCH EQUILIBRIUM By Giuseppe Moscarini and Fabien Postel-Vinay February 2010 COWLES FOUNDATION DISCUSSION PAPER NO. 1754 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/ Stochastic Search Equilibrium∗ Giuseppe Moscarini† Fabien Postel-Vinay‡ Yale University University of Bristol and and NBER Paris School of Economics February 2010 Abstract We study a stochastic economy where both employed and unemployed workers search randomly for labor contracts posted by firms, while aggregate productivity is subject to persistent shocks. Our exercise provides the first dynamic stochastic gen- eral equilibrium analysis of a popular class of search wage-posting models, drawing in part from the literature on recursive contracts under moral hazard. Each firm offers and commits to a (Markov) contract, which specifies a wage contingent on all payoff- relevant states, but must pay equally all of its workers, who have limited commitment and are free to quit at any time. An equilibrium of this contract-posting game is Rank- Preserving [RP] if larger firms offer a larger value to their workers in all states of the world. We find two sufficient (but not necessary) conditions for every equilibrium to be RP: either firms only differ in their initial size, or they also differ in their fixed id- iosyncratic productivity but more productive firms are initially weakly larger, in which case turnover is always efficient, as workers always move from less to more productive firms. In both cases, the ranking of firm sizes never changes on the RP equilibrium path, a property that has three useful implications. First, the stochastic dynamics of firm size, uniquely pinned down in equilibrium, provide an intuitive explanation for the empirical finding that large employers are more cyclically sensitive (Moscarini and Postel-Vinay, 2009). Second, contracts are unique in RP equilibrium. Third, RP equi- librium computation is tractable, and we construct and simulate calibrated examples. Keywords: Equilibrium Job Search, Dynamic Contracts, Stochastic Dynamics. JEL codes: J64, J31, D86. ∗Earlier versions of this paper circulated under the title “Non-stationary search equilibrium”. We acknowl- edge useful comments to earlier drafts of this paper from seminar and conference audiences at numerous venues. We also wish to thank Ken Burdett, Dale Mortensen and Rob Shimer for constructive discussions of earlier versions of this paper. The usual disclaimer applies. †Address: Department of Economics, Yale University, PO Box 208268, New Haven CT 06520-8268. Tel. +1-203-432-3596. E-mail [email protected]. Web http://www.econ.yale.edu/faculty1/moscarini.htm ‡Address: Department of Economics, University of Bristol, 8 Woodland Road, Bristol BS8 1TN, UK. Tel: +44 117 928 8431. E-mail [email protected]. Web www.efm.bris.ac.uk/economics/staff/vinay/. Postel-Vinay is also affiliated with CEPR (London) and IZA (Bonn). 1 Introduction We study the equilibrium dynamics of a frictional labor market where firms offer and commit to employment contracts and workers search randomly on and off the job for those contracts, while aggregate productivity is subject to persistent shocks. In a broad sense, our exercise sheds light on the long-term contracts that emerge in a market equilibrium environment, in the presence of moral hazard and aggregate uncertainty. More specifically, we perform the first analysis of aggregate stochastic dynamics in a popular class of search wage-posting models, originating with Burdett and Mortensen (1998, henceforth BM). By providing a coherent formalization of the hypothesis that cross-sectional wage dispersion is largely a consequence of labor market frictions, the BM model has started a fruitful line of research in the analysis of wage inequality and worker turnover, as the vibrant and empirically very successful literature organized around that hypothesis continues to show (see Mortensen, 2003 for an overview). That literature, however, is invariably cast in deterministic steady state. Ever since the first formulation of the BM model, job search scholars have regarded the characterization of its out-of-steady-state behavior as a daunting problem, essentially because one of the model’s state variables, which is also the main object of interest, is the endogenous distribution of wage (or job value) offers. This is an infinite-dimensional object, endogenously determined in equilibrium as the distribution across firms of offer strategies that are mutual best responses, and evolving stochastically with the aggregate impulse. In this paper, we solve this problem. We show that, under mild sufficient conditions, the economy under investigation has a unique equilibrium. In this equilibrium, the workers’ ranking of firms is the same in all aggregate states — what we call a Rank-Preserving Equilibrium (RPE). The sufficient conditions are that firms either are equally productive, or differ in the permanent component of their productivity and the more productive they are, the (weakly) more workers they initially employ — for example, they all start empty. In the latter case, in RPE more productive firms offer a larger value and employ more workers at all points in time: when given a chance, a worker always moves from a less into a more productive firm, so that the equilibrium allocation of employment is constrained efficient. This parallels a similar property of BM’s static equilibrium. To illustrate and qualify our contributions we now provide details about the economy that we study, our solution method, and the nature of the unique equilibrium. Infinitely lived and risk neutral firms and workers come in contact infrequently. Firms produce homogenous output with labor in a linear technology, which may permanently differ across firms. Aggre- gate multiplicative TFP shocks affect labor productivity as well as the job contact rates, on 1 and off the job, the exogenous job destruction rate, and the value of leisure. In this economy, the constrained efficient allocation is easily characterized. When given the opportunity, an employed worker is moved from a less productive to a weakly more productive firm. We abstract from issues of entry and exit and restrict attention to parameter configurations such that, as in BM, employment is always preferred to unemployment. This efficient turnover behavior describes a simple Markov process for the evolution of the firm size distribution. If we shut down aggregate shocks, from any initial condition which gives rise to RPE this process converges deterministically to the size distribution that BM found by solving directly for the stationary distribution. In a stochastic environment, for any history of aggregate shocks, we can solve in closed form for the path of the socially optimal distribution of employment across firms, thus of the size of each firm type. The next step is equilibrium analysis. We assume that firms offer and commit to a contract which conditions the wage on all possible relevant states and is subject to an equal- treatment constraint: it must pay the same wage in a given period to all of its employees, whether incumbent, newly hired from unemployment or from employment. This constraint indeed defines the boundaries of a firm. Workers cannot commit not to quit to other jobs when the opportunity arises, or to unemployment whenever they please, so commitment is one-sided and firms face a standard moral hazard problem. Contract offers are privately observed only by the recipients, thus deviations cannot be detected by other players. We look for a Sequential Nash equilibrium of this contract-posting game. We find the largest state space on which equilibrium contracts can be conditioned. For tractability, we then restrict attention to a Markov Perfect equilibrium, where wages depend only on payoff-relevant states: two exogenous, the productivity of the firm and the state of aggregate productivity, one endogenous to the firm, its current size, and one endogenous to the economy but exogenous to the firm, the distribution of employment across all firms. A firm must track this infinitely-dimensional object in order to know the distributions of competing offers and of values earned by currently employed workers, thus how much recruitment and retention its own contract will generate. Equilibrium only imposes two very weak restrictions on these two distributions: they must have no atoms and a connected support. Yet we are able to establish that at most one Markov perfect equilibrium exists, characterize that unique equilibrium and show that it decentralizes the constrained efficient allocation. The key step in our analysis is a comparative dynamics property of the best-response. We show that, at any node in the game and for any distribution of offers made by other firms and of values earned by employed workers, a more productive and/or larger firm optimally offers a contract that pays its existing and new workers a larger value. Therefore, if firms are homogeneous, or if more productive firms are initially no smaller, then no firm wants 2 to break ranks in the distribution of competing offers, which then coincides with the given distribution of firm productivities or initial sizes. This immediately implies our main result that equilibrium, if it exists, is unique and RP, thus constrained efficient. The intuition behind this comparative dynamics property of the best-response contract parallels a single-crossing property of the static BM model. There, a more productive firm gains more from employing a worker, hence wants to (and can) pay a higher wage. In addi- tion, under the equal treatment constraint, the effect of the wage on retention is proportional to own size, while that on hiring is independent of size. Finally, size increases in the wage due to its effect on recruitment and retention. Thus, ceteris paribus, a larger firm also wants to pay more. This intuition does not extend immediately to our dynamic stochastic setting, because firm size is an evolving state variable with a given initial condition.
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